If `alpha, beta, gamma` are the zeros of the cubic polynomial `ax^3+bx^2+cx+d`, then `alphabeta+betagamma+gammaalpha` is equal to:

Options

A. `-b/a`
B. `b/a`
C. `c/a`
D. `d/a`

Answer

To find the value of \( \alpha\beta + \beta\gamma + \gamma\alpha \) for the cubic polynomial \( ax^3 + bx^2 + cx + d \) where \( \alpha, \beta, \gamma \) are the zeros, we can use Vieta's formulas.

### Step-by-Step Solution:

1. **Identify the coefficients of the polynomial**:
The given cubic polynomial is \( ax^3 + bx^2 + cx + d \). Here, the coefficients are:
- \( A = a \) (coefficient of \( x^3 \))
- \( B = b \) (coefficient of \( x^2 \))
- \( C = c \) (coefficient of \( x \))
- \( D = d \) (constant term)

2. **Use Vieta's formulas**:
According to Vieta's formulas for a cubic polynomial, the relationships between the roots (zeros) and the coefficients are:
- \( \alpha + \beta + \gamma = -\frac{B}{A} \)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{C}{A} \)
- \( \alpha\beta\gamma = -\frac{D}{A} \)

3. **Find \( \alpha\beta + \beta\gamma + \gamma\alpha \)**:
From Vieta's formulas, we directly have:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{C}{A} \]

4. **Conclusion**:
Therefore, the value of \( \alpha\beta + \beta\gamma + \gamma\alpha \) is equal to \( \frac{c}{a} \).

### Final Answer:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]